The American Civil Liberties Union (ACLU), Electronic Frontier Foundation (EFF) and Free Press have pressed the public to contact their senators this week about debating these amendments. The groups charge that the surveillance measure lacks transparency and could be used to sweep up American citizens' communications without a warrant.

In a blog post published on Wednesday, EFF called for more transparency about how the law works.

Michelle Richardson, a legislative counsel in the ACLU's Washington office, also said the Senate should take some time to debate a handful of amendments to the bill rather than just passing a clean five-year extension of it.

Richardson said voting on the amendments would be an "opportunity to make even the most modest of changes to one of the most sweeping surveillance laws passed since 9/11."

The bill is set to expire on Dec. 31. The House voted in favor of reauthorizing the measure in September without adding any amendments—which GOP senators hope to do as well.

Merkley plans to offer an amendment that would require the government to declassify the Foreign Intelligence Surveillance Court's opinions on surveillance requests. Wyden hopes to offer an amendment that would prevent intelligence officials from searching through the communications they've collected under the surveillance law for emails or phone calls of specific Americans.